$\dfrac{ 8e - 5f }{ -9 } = \dfrac{ 9e - 6g }{ 4 }$ Solve for $e$.
Explanation: Multiply both sides by the left denominator. $\dfrac{ 8e - 5f }{ -{9} } = \dfrac{ 9e - 6g }{ 4 }$ $-{9} \cdot \dfrac{ 8e - 5f }{ -{9} } = -{9} \cdot \dfrac{ 9e - 6g }{ 4 }$ $8e - 5f = -{9} \cdot \dfrac { 9e - 6g }{ 4 }$ Multiply both sides by the right denominator. $8e - 5f = -9 \cdot \dfrac{ 9e - 6g }{ {4} }$ ${4} \cdot \left( 8e - 5f \right) = {4} \cdot -9 \cdot \dfrac{ 9e - 6g }{ {4} }$ ${4} \cdot \left( 8e - 5f \right) = -9 \cdot \left( 9e - 6g \right)$ Distribute both sides ${4} \cdot \left( 8e - 5f \right) = -{9} \cdot \left( 9e - 6g \right)$ ${32}e - {20}f = -{81}e + {54}g$ Combine $e$ terms on the left. ${32e} - 20f = -{81e} + 54g$ ${113e} - 20f = 54g$ Move the $f$ term to the right. $113e - {20f} = 54g$ $113e = 54g + {20f}$ Isolate $e$ by dividing both sides by its coefficient. ${113}e = 54g + 20f$ $e = \dfrac{ 54g + 20f }{ {113} }$